3.32.39 \(\int \frac {-2 e^x-x}{-1+e^x+x} \, dx\)

Optimal. Leaf size=18 \[ 14-x-\log \left (1-e^x-x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 e^x-x}{-1+e^x+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*E^x - x)/(-1 + E^x + x),x]

[Out]

-2*x - 2*Defer[Int][(-1 + E^x + x)^(-1), x] + Defer[Int][x/(-1 + E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-2+\frac {-2+x}{-1+e^x+x}\right ) \, dx\\ &=-2 x+\int \frac {-2+x}{-1+e^x+x} \, dx\\ &=-2 x+\int \left (-\frac {2}{-1+e^x+x}+\frac {x}{-1+e^x+x}\right ) \, dx\\ &=-2 x-2 \int \frac {1}{-1+e^x+x} \, dx+\int \frac {x}{-1+e^x+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 17, normalized size = 0.94 \begin {gather*} -x-\log \left (1-e^x-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^x - x)/(-1 + E^x + x),x]

[Out]

-x - Log[1 - E^x - x]

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 12, normalized size = 0.67 \begin {gather*} -x - \log \left (x + e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)-x)/(x+exp(x)-1),x, algorithm="fricas")

[Out]

-x - log(x + e^x - 1)

________________________________________________________________________________________

giac [A]  time = 0.31, size = 12, normalized size = 0.67 \begin {gather*} -x - \log \left (x + e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)-x)/(x+exp(x)-1),x, algorithm="giac")

[Out]

-x - log(x + e^x - 1)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 13, normalized size = 0.72

method	result	size
norman	\(-x -\ln \left (x +{\mathrm e}^{x}-1\right )\)	\(13\)
risch	\(-x -\ln \left (x +{\mathrm e}^{x}-1\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*exp(x)-x)/(x+exp(x)-1),x,method=_RETURNVERBOSE)

[Out]

-x-ln(x+exp(x)-1)

________________________________________________________________________________________

maxima [A]  time = 0.75, size = 12, normalized size = 0.67 \begin {gather*} -x - \log \left (x + e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)-x)/(x+exp(x)-1),x, algorithm="maxima")

[Out]

-x - log(x + e^x - 1)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 12, normalized size = 0.67 \begin {gather*} -x-\ln \left (x+{\mathrm {e}}^x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + 2*exp(x))/(x + exp(x) - 1),x)

[Out]

- x - log(x + exp(x) - 1)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 10, normalized size = 0.56 \begin {gather*} - x - \log {\left (x + e^{x} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*exp(x)-x)/(x+exp(x)-1),x)

[Out]

-x - log(x + exp(x) - 1)

________________________________________________________________________________________